Tests were performed to simulate the practical problem of determining the optical density of a neutral density filter using Moon observations.

Experiment 1: A real image of the Moon was divided by 100.0 over half the illuminated disc, resulting in an image that resembled what an image of the Moon seen through a focal-place ND filter would look like. The image was one that Tom Stone of the USGS donated, from the ROLO project. A synthetic image of the Moon for the time of observation was then generated using the DMI lunar simulator. Two types of images were generated – one using the Lambertian BDRF and one using Hapke’s BRDF. The synthetic image was then rotated and scaled so that it aligned with the real image of the Moon. This procedure included the application of a multiplicative factor over that part of the image which coincided with the ‘filtered’ part of the observed image. The factor was one of in total 6 parameters that were determined, using least squares techniques by minimising the residual between the ‘observed’ image and the synthetic image. Results: The values found for the ‘filter factor’ in the minimisation were 99.57 and 100.332 for the Lambertian image and the Hapke image, respectively. Since the actual factor was 100 we see that we are able to determine the filter OD to within 0.6 and 0.3 %. A second experiment employed a factor of 1000 and we determined 999.57 and 1005.9 – i.e. 0.04% and 0.6%. Notes on the procedures used: The minimisation method requires a good starting guess for all parameters – i.e. image shifts, image scaling, image rotation, image intensity and the factor for the OD of the simulated filter. This starting guess was obtained by first performing the minimisations to align and intensity-scale the images and then fixing these and determining the OD factor and finally letting all parameters be determined from the previous best guesses. To align the images only the image half that was not ‘under the filter’ was used. The residuals’ RMSE is very sensitive to small shifts in the image placements, due to the amount of detail in the images of the lunar disc – small shifts soon place dark areas over bright ones. It is speculated that smoothing of both images before the above procedure is applied could solve this problem. This is analogous to suggesting that the Moon is not observed in focus or at least not in perfect focus. The above was not performed with realistic Poisson noise on the observed filtered half, nor anywhere in the synthetic image. Scattered light from the lunar disc is not modelled in the synthetic image, but could be. Use of the residual RMSE as target of minimisation may be a bad idea – the numbers are small on the bright side of the filter and low on the dark side: Any contribution to the RMSE from the dark side should be weighted higher. Perhaps RMSE of the residual relative to the observed image should be minimised.

Experiment 2: Realistic Poisson noise was included in the ‘observed’ image on the dark filtered side. At first a filter OD of 10 was used, and minimisation recovered 10.04492 and 10.045737 – i.e. 0.4% and 0.5% errors. For larger filter factors initial results were dismaying: a large bias was apparent in the rsults – must consider again.